Learn Italian: La ragazza chiede una cosa importante. - The girl asks an important thing.

Because it comes from the regular -ere verb chiedere (to ask). To form the he/she/it form (matching la ragazza), you take off the -ere ending and add -e.

Is chiedere used for asking questions or asking for things?

Chiedere means to ask in the sense of making a request or asking for something, which fits here. If you want to say to ask a question, Italians usually use the phrase fare una domanda (literally to make a question).

Cosa is a feminine word, so why doesn't the adjective end in -a (importanta)?

Some Italian adjectives, like importante, end in -e in their dictionary form. These adjectives do not change between masculine and feminine; they use the same -e ending for all singular nouns. They only change their ending to -i when they become plural.

Why does importante come after una cosa?

In English we put the describing word before the noun (an important thing), but in Italian, most adjectives come directly after the noun they describe.